Use of 'gravitational lensing' to independently measure the mass of galaxies.
In one sense, Einstein was lucky: very soon after he published the general theory of relativity (GR), one of its predictions was confirmed ... the deflection of light by the mass of the Sun. Of course, GR was already on a sound observational basis, through its post-diction (explanation) of the advance of the perihelion of Mercury, which had been known for many decades. It also helped that Eddington was an enthusiastic supporter, so the observations of stars near the limb of the (eclipsed) Sun were interpreted as confirmation (today we'd call those observations marginal).
It was soon determined that a massive, compact body such as a galaxy should be able to 'lens' a more distant object, such as another galaxy; it was also quickly realised that this would happen only very rarely.
However,
in 1979 just such an example of (strong) gravitational lensing was discovered, the distant object being a quasar.
With the angular resolution of the Hubble Space Telescope (HST), many examples of such strong lensing can be discovered;
a very recent HST PR gives some examples, and the associated papers and articles explain how many more will likely be discovered in the next few decades.
Estimating the mass of the 'lens' is relatively straight-forward, using textbook physics (GR), although in practice there are complications.
The beauty of this technique is that it is completely independent of the others I've discussed so far: the only thing that matters for GR is the total mass.
And while the number of galaxies whose masses have been estimated using this technique is, today, still small, the results are consistent with those from the other techniques.
There is another kind of gravitational (due to GR) lensing, 'weak lensing', also called 'shear'. In this case, the shape of a distant object is distorted, but only subtly. So for this technique to work, you need lots and lots of distant objects whose undistorted shapes are known. Fortunately, this is exactly what some of the recent large surveys can do, though in this case you get an averaged picture of the mass distribution of a lot of galaxies, rather than of just one.
One possible result from weak lensing observations, of galaxies, is an estimate of the shape of the mass distribution in the halos; the other techniques can give only estimates of the 'mass within distance x' - i.e. the distribution of mass assuming it is spherical (detailed studies of MW halo stars are an exception, as are some studies of some dwarf galaxies; however these are too new to say much about yet, and in any case would apply to only one, or a very few, galaxies).
Weak lensing observations of galaxies is relatively new, however results (estimates of galaxy halo mass) to date are consistent with those from techniques, and there are some early results on the shape of those halos (e.g. there is some 'flattening' and 'ellipticity'), on how they differ from one type of galaxy to another (e.g. giant elliptical galaxies in the centres of rich clusters seem to have different halos than other galaxies), and on how they depend on galaxy environment (e.g. galaxy halos in rich clusters seem to be 'truncated').
Next: so how do we know that this mass, in galaxy (halos), which is not emitting or absorbing light, is CDM?
